Find the radius of the middle circle

In summary: The hypotenuse of the green triangle is 8+x, its vertical leg is x‒8 .The hypotenuse of the orange triangle is 4+x, its vertical leg is x‒4 .In summary, the radius of the middle circle is 8.
  • #1
xcrunner448
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Homework Statement



There are 3 circles, each tangent to 2 lines and to each other (as in the picture). The radius of the right (largest) circle is 8, and the radius of the left (smallest) circle is 4. What is the radius of the middle circle?


The Attempt at a Solution



I tried using similar triangles to solve the problem, using the right triangles formed by the bottom line, the radii of the circles, and the angle bisector (which goes through the centers of all of the circles). However, I could not find any relation that would give me an equation to find x. I also tried to somehow calculate the angle between the two lines, but that got really complicated. I am not sure what other method I could use to solve this, and any help would be appreciated.
 

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  • #2
Can you show us? From the graph, we can at least see three equal ratios. Let the distance from the vertex to the first circle be y. Working on from there get the three equal ratios from which you should be able to get two equations. You can always eliminate y and solve for x.
 
  • #3
Using similar triangles should get you the answer (hint: you have to solve two equations with two unknowns).
 
  • #4
Ok I think I got it now. Using y as the distance from the vertex to the center of the first circle, I got y/4 = (4+x)/x and y-8-x = 4+x. From the second equation I got y=2x+12, and plugging that into the first equation I found that x=4*sqrt(2). I guess I just didn't see that when I first tried to solve it. Thank you!

In the meantime I think I solved it another way, although it is much more complex. After quite a bit of work I found that the angle between the two tangent lines for tangential circles of radii a and b (with b>a) is 2*arcsin((b-a)/(a+b)). I did this for each pair of circles (b=8, a=x and b=x, a=4) and set them equal to each other, giving 2*arcsin((8-x)/(8+x))=2*arcsin((x-4)/(x+4)), which led to the proportion (8-x)/(8+x)=(x-4)/(x+4). When I solved that I got x=4*sqrt(2), the same answer as above. It was a lot more work than the similar triangles method, but it worked. Also, I thought it was interesting that the radius of the middle circle is always the geometric mean of the radii of the two outer circles. (x=4*sqrt(2)=sqrt(32)=sqrt(4*8))
 
  • #5
You can do it directly with the right triangles indicated on the figure I modified using your figure.
The hypotenuse of the red triangle is x+4, its vertical leg is x‒4 .
The hypotenuse of the blue triangle is x+8, its vertical leg is 8‒x .
 

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What is the middle circle in a scientific experiment?

The middle circle in a scientific experiment refers to the circle at the center of a larger set of circles, typically used to represent data or observations.

Why is it important to find the radius of the middle circle?

Finding the radius of the middle circle can provide crucial information about the data being observed. It can help determine the distribution of data points and identify any patterns or trends present.

How can the radius of the middle circle be calculated?

The radius of the middle circle can be calculated by measuring the distance from the center of the circle to any point on its edge. This distance is known as the radius and can be measured using a ruler or other measuring tool.

What factors can affect the radius of the middle circle?

The radius of the middle circle can be affected by various factors such as the size of the data set, the scale of the graph or chart, and the precision of the measurements taken. Additionally, any errors or outliers in the data can also impact the radius.

How can the radius of the middle circle be used in data analysis?

The radius of the middle circle can be used to calculate other important measures such as the diameter, circumference, and area of the circle. It can also be used to compare the data points within the circle and make inferences about the overall data set.

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